Projective curve/Vector bundle/Cohomology class/Finite annihilation and torsor/Fact/Proof

Proof

If (1) holds, then the pull-back ${}\varphi ^{*}(T)=T\times _{C}C'$ is trivial (as a torsor), as it equals the torsor given by ${}\varphi ^{*}(c)=0$. Hence ${}\varphi ^{*}(T)$ is isomorphic to a vector bundle and contains in particular a copy of ${}C'$. The image ${}Z$ of this copy is a projective curve inside ${}T$.

If (2) holds, then let ${}C'$ be the normalization of ${}Z$. Since ${}Z$ dominates ${}C$, the resulting morphism

$\varphi \colon C'\longrightarrow C$

is finite. Since this morphism factors through ${}T$ and since ${}T$ annihilates the cohomology class by which it is defined, it follows that

${}\varphi ^{*}(c)=0$.

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